Elementary Mathematics Compound interest formulas Exponential and logarithmic functions This is an article from my home page: www.olewitthansen.dk Ole Witt-Hansen 2011 Contents Chapter 1. Compound interest .............................................................................................................1 1. Percentage calculations................................................................................................................1 2. The compound interest formula ...................................................................................................2 3. The average rate of growth ..........................................................................................................4 Chapter 2. Savings annuities and debts annuities ...............................................................................5 1. Annuities ..................................................................................................................................5 1.1 Formula for the sum of a geometric series.................................................................................5 1.2 Savings annuity..........................................................................................................................6 1.3 Debts annuities.......................................................................................................................6 Chapter 3. Exponential functions........................................................................................................8 1. The concept of a power of a number............................................................................................8 1.1 Calculation with powers.............................................................................................................8 2. Extension of powers to negative exponents.................................................................................9 3. Extension of powers to rational number exponents...................................................................10 Chapter 4. Logarithmic functions. .....................................................................................................14 1. Logarithmic functions............................................................................................................14 Chapter 5. Eksponential growth and power growth...........................................................................17 1. Exponential growth................................................................................................................17 1.1 Solution of exponential equations........................................................................................17 2. Doubling constant and half-live constant...............................................................................18 3. Logaritmic scale.....................................................................................................................19 4. Power functions.........................................................................................................................21 Percentage and interest 1 Chapter 1. Compound interest 1. Percentage calculations The well known percent symbol “%” has been invented to signify 1/100. If we want find 3.5% of 450, then it usually done in the following way (in the grammar school) 1% of 450 is 450/100 = 4.50. 3.5% of 450 = 3.54.50 =15.75. But since 3.5% = 0.035, so that 3.5% of 450 can be directly calculated as 4500.035 = 15.75 From now on we shall always use the short method to calculate a percentages, also for that reason that it is awkward to have units and special symbols in mathematical equations. Often the letter p is used as the symbol for a certain percentage. p/100 is then denoted r. Thus we have the connection: r = p% = p/100. r is called the interest rate or the growth rate. When we shall calculate the interest R from the capital K, and the interest rate is r, then we have according to the example above: (1.1) R = Kr This formula is of course equally valid, even if it is does not capital capital, but common calculation with percentages. If we want to find p% = r from a quantity k, and the result is i (interest), then we have an equivalent formula: (1.2) i = kr However the formula is easier to remember and read, if we keep the letters K, R and r, even if it is not about money so the concepts capital, Interest, and interest rate is to be understood in a broader sense. Using this (mathematical) formula, we may for example answer a question like: How many percent make 27 up of 179. Inserting R = 27 and K = 179 in the formula (1.1) and solving for r. we find: R 27 R Kr r 0.1508 15.08% K 179 We shall now derive the very important formula, which expresses how much a capital K (or any other quantity having percentage growth) has grown to, when it has increased by r = p% . If the accumulated capital is K1, then K1 is equal to K plus the interest R = Kr. (1.3) K1 = K + R = K + Kr = K(1+r), so: K1 = K(1 + r) Percentage and interest 2 Notice that the formula is equally valid, if we have a percentage decrease instead of a growth. It just means that the interest rate r is negative. That a quantity decreases with 15% is the same as a growth of -15%. Eample: 1. A department store advertises that the price is reduced by 20 % for a certain product. The product now costs 477.- What was the price before the reduction? We apply the formula (1.5) with K1 = 457, r = - 20% = - 0.2 to determine K. K = 457/(1-0.2) = 571.25. 2. An enterprise had one year an income of 257.000,- and expenses were 301.000. The following year the income grew with 5%, while the expenses increased by 3%. How many percent has the deficit grown/decreased? If we set the deficit for the two years to u and u1, we have: u = 257.000 - 301.000 = - 44.000 and u1 = 257.0001.05 - 301.0001.03 = - 40.180 We then apply (1.3): 40.180 = 44.000(1+r) (1+r) = 40180/44000 = 0.9132 then r = 0.9132-1 = -0.0868. Which means that the deficit has been reduced by 8.68 %. 2. The compound interest formula When a capital accumulates interest in a banking institution, the interest is attributable, with a fixed interval called the term. The period between two attributable of interest may be one year, half a year or monthly or whatever has been agreed. But when you talk about the interest rate of a deposit or a loan, the interest rate is always indicated as the yearly interest rate, even if the term is shorter. If a bank makes a loan to 5% p.a. (pro annum = yearly), and the interest rates are due each half - year, the semi-annually interest rate is 2.5%. As we shall show, however, the effective interest rate will be slightly larger than 5 %. If the capital K has been repaid with the interest rate r in n terms, one might think that the accumulated interest was nKr (n times the interest in one term). This is, however, not the case, since the capital has grown after each term, and therefore the amount due to collect interest has grown. You will also receive interest from the interest. We shall then establish a formula for Kn , the accumulated capital after n terms when the start capital is K and the interest rate is r. We know already from (1.3) that we can find the capital after one term by multiplying by (1+r). From this follows: K1 = K(1+r) (The capital after one term) 2 K2 = K1(1+r) = K(1+r) (The capital after two terms) 3 K3 = K2(1+r) = K(1+r) (The capital after three terms) ........... n Kn = Kn-1(1+r) = K(1+r) (The capital after n terms) We then find the very important compound interest formula. Percentage and interest 3 n (1.4) Kn = K(1+r) Kn is the accumulated capital, when the capital has gained interest in n terms with an interest rate r. The compound interest formula is by no way limited to capital, but may be applied to any quantity which has a constant percentage growth. If a quantity b has a percentage rate in one period it has grown to bn,, after n-periods where: n (1.5) bn =b(1+r) If this is perceived as a function, we may write: f(n) = b(1+r)n. Traditionally we put: a = 1+r. a may then be called the projection factor. Thus we may write: f(n) = b(1+r)n f(n) = ban where a = 1+r r = a - 1 The interface between the compound interest formula and the exponential functions f(x) = b ax should then be obvious, since we only have to replace the integer variable n with the real variable x. 2.1 Eksempel. 1. A deposit in a bank having an interest rate of 4.5% p.a is 2.000. The term length is 6 months. Find the deposit after 7 years. The number of terms are then n =14. The interest rate is r = 0.0225 and K = 2000. When we insert in (2.1), we find: 14 K14 = 2000 (1,0225) = 2.730.97 2. At a purchase on installment, the buyer is offered a loan having monthly terms at an interest rate of 1.5%. You may think that this should amount to: 12∙ 1.5% = 18% per annum? Calculate the true effective annual interest. We apply (2.1) with k = 1, to se how much 1 will grow to in 12 terms with an interest rate of 1.5% =0.015. 12 k12 = (1.015) = 1.1956 Which correspond to a yearly interest rate of 19.56%. Because of the interest rate on the interest, the effective interest rate in n terms will be somewhat larger than n times the interest rate in 1 term. 3. The stock of herring in the Baltic see in mill. tons, could since 1987 be described by an exponential function f(x) = 132(0,82)x, where x is the number of years passed since 1987. Determine the yearly rate of growth. a = 0.82, so r = 0.82-1 = -0.18 = - 18%. The stock of herring in the Baltic see has thus